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Chapter 2 | Limits
Figure 2.33 The function f ( x ) is not continuous at a because lim x → a f ( x ) does not exist.
However, as we see in Figure 2.34 , these two conditions by themselves do not guarantee continuity at a point. The function in this figure satisfies both of our first two conditions, but is still not continuous at a . We must add a third condition to our list: iii. lim x → a f ( x ) = f ( a ).
Figure 2.34 The function f ( x ) is not continuous at a because lim x → a f ( x ) ≠ f ( a ).
Now we put our list of conditions together and form a definition of continuity at a point.
Definition A function f ( x ) is continuous at a point a if and only if the following three conditions are satisfied: i. f ( a ) is defined ii. lim x → a f ( x ) exists iii. lim x → a f ( x ) = f ( a ) A function is discontinuous at a point a if it fails to be continuous at a .
The following procedure can be used to analyze the continuity of a function at a point using this definition.
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